Multiplicity of Positive Solutions for a Quasilinear Schrödinger Equation with an Almost Critical Nonlinearity

Author:

Figueiredo Giovany M.1,Severo Uberlandio B.2,Siciliano Gaetano3ORCID

Affiliation:

1. Departamento de Matemática , Universidade de Brasília , 70910-900 , Brasília , DF , Brazil

2. Departamento de Matemática , Universidade Federal da Paraíba , 58051-900 , João Pessoa , PB , Brazil

3. Departamento de Matemática , Instituto de Matemática e Estatística , Universidade de São Paulo , Rua do Matão 1010, 05508-090 São Paulo , SP , Brazil

Abstract

Abstract In this paper we prove an existence result of multiple positive solutions for the following quasilinear problem: { - Δ u - Δ ( u 2 ) u = | u | p - 2 u in  Ω , u = 0 on  Ω , \left\{\begin{aligned} \displaystyle-\Delta u-\Delta(u^{2})u&\displaystyle=|u|% ^{p-2}u&&\displaystyle\phantom{}\text{in }\Omega,\\ \displaystyle u&\displaystyle=0&&\displaystyle\phantom{}\text{on }\partial% \Omega,\end{aligned}\right. where Ω is a smooth and bounded domain in N , N 3 {\mathbb{R}^{N},N\geq 3} . More specifically we prove that, for p near the critical exponent 22 * = 4 N / ( N - 2 ) {22^{*}=4N/(N-2)} , the number of positive solutions is estimated below by topological invariants of the domain Ω: the Ljusternick–Schnirelmann category and the Poincaré polynomial. With respect to the case involving semilinear equations, many difficulties appear here and the classical procedure does not apply immediately. We obtain also en passant some new results concerning the critical case.

Publisher

Walter de Gruyter GmbH

Subject

General Mathematics,Statistical and Nonlinear Physics

Cited by 1 articles. 订阅此论文施引文献 订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献

1. The effect of the domain topology on the number of positive solutions for an elliptic system;Partial Differential Equations and Applications;2022-09-20

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